LMFDB - Embedded newform 320.3.e.a.159.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(159,320)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("320.159");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.71936845953$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.691798081536.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 $ x^8 - 4x^7 + 26x^6 - 64x^5 + 229x^4 - 356x^3 + 164x^2 + 4*x + 985
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			159.7
Root			$1.72474 + 0.954705i$ of defining polynomial
Character	$\chi$	$=$	320.159
Dual form			320.3.e.a.159.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{3} +(2.44949 - 4.35890i) q^{5} -10.6771 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+2.44949i q^{3} +(2.44949 - 4.35890i) q^{5} -10.6771 q^{7} +3.00000 q^{9} -8.71780 q^{11} -19.5959 q^{13} +(10.6771 + 6.00000i) q^{15} +21.3542i q^{17} -26.1534 q^{19} -26.1534i q^{21} +10.6771 q^{23} +(-13.0000 - 21.3542i) q^{25} +29.3939i q^{27} -34.8712i q^{29} +4.00000i q^{31} -21.3542i q^{33} +(-26.1534 + 46.5403i) q^{35} -14.6969 q^{37} -48.0000i q^{39} -24.0000 q^{41} -56.3383i q^{43} +(7.34847 - 13.0767i) q^{45} +10.6771 q^{47} +65.0000 q^{49} -52.3068 q^{51} -48.9898 q^{53} +(-21.3542 + 38.0000i) q^{55} -64.0625i q^{57} +43.5890 q^{59} +26.1534i q^{61} -32.0312 q^{63} +(-48.0000 + 85.4166i) q^{65} +7.34847i q^{67} +26.1534i q^{69} +84.0000i q^{71} +106.771i q^{73} +(52.3068 - 31.8434i) q^{75} +93.0806 q^{77} -100.000i q^{79} -45.0000 q^{81} +17.1464i q^{83} +(93.0806 + 52.3068i) q^{85} +85.4166 q^{87} +150.000 q^{89} +209.227 q^{91} -9.79796 q^{93} +(-64.0625 + 114.000i) q^{95} -21.3542i q^{97} -26.1534 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 104 q^{25} - 192 q^{41} + 520 q^{49} - 384 q^{65} - 360 q^{81} + 1200 q^{89}+O(q^{100}) $ 8 * q + 24 * q^9 - 104 * q^25 - 192 * q^41 + 520 * q^49 - 384 * q^65 - 360 * q^81 + 1200 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			2.44949i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			2.44949i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$	2.44949	−	4.35890i	0.489898	−	0.871780i
$5$	2.44949	−	4.35890i	0.489898	−	0.871780i
$6$		0			0
$7$	−10.6771			−1.52530			−0.762648	−	0.646813i	$-0.776101\pi$
$7$	−10.6771			−1.52530			−0.762648	+	0.646813i	$0.776101\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	−8.71780			−0.792527			−0.396264	−	0.918137i	$-0.629693\pi$
$11$	−8.71780			−0.792527			−0.396264	+	0.918137i	$0.629693\pi$
$12$		0			0
$13$	−19.5959			−1.50738			−0.753689	−	0.657231i	$-0.771728\pi$
$13$	−19.5959			−1.50738			−0.753689	+	0.657231i	$0.771728\pi$
$14$		0			0
$15$	10.6771	+	6.00000i	0.711805	+	0.400000i
$16$		0			0
$17$			21.3542i			1.25613i	0.778162	+	0.628063i	$0.216152\pi$
$17$			21.3542i			1.25613i	−0.778162	+	0.628063i	$0.783848\pi$
$18$		0			0
$19$	−26.1534			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−26.1534			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$		−	26.1534i		−	1.24540i
$22$		0			0
$23$	10.6771			0.464221			0.232110	−	0.972689i	$-0.425437\pi$
$23$	10.6771			0.464221			0.232110	+	0.972689i	$0.425437\pi$
$24$		0			0
$25$	−13.0000	−	21.3542i	−0.520000	−	0.854166i
$26$		0			0
$27$			29.3939i			1.08866i
$28$		0			0
$29$		−	34.8712i		−	1.20245i	−0.799078	−	0.601227i	$-0.794679\pi$
$29$		−	34.8712i		−	1.20245i	0.799078	−	0.601227i	$-0.205321\pi$
$30$		0			0
$31$			4.00000i			0.129032i	0.997917	+	0.0645161i	$0.0205504\pi$
$31$			4.00000i			0.129032i	−0.997917	+	0.0645161i	$0.979450\pi$
$32$		0			0
$33$		−	21.3542i		−	0.647096i
$34$		0			0
$35$	−26.1534	+	46.5403i	−0.747240	+	1.32972i
$36$		0			0
$37$	−14.6969			−0.397215			−0.198607	−	0.980079i	$-0.563642\pi$
$37$	−14.6969			−0.397215			−0.198607	+	0.980079i	$0.563642\pi$
$38$		0			0
$39$		−	48.0000i		−	1.23077i
$40$		0			0
$41$	−24.0000			−0.585366			−0.292683	−	0.956210i	$-0.594548\pi$
$41$	−24.0000			−0.585366			−0.292683	+	0.956210i	$0.594548\pi$
$42$		0			0
$43$		−	56.3383i		−	1.31019i	−0.755546	−	0.655096i	$-0.772628\pi$
$43$		−	56.3383i		−	1.31019i	0.755546	−	0.655096i	$-0.227372\pi$
$44$		0			0
$45$	7.34847	−	13.0767i	0.163299	−	0.290593i
$46$		0			0
$47$	10.6771			0.227172			0.113586	−	0.993528i	$-0.463766\pi$
$47$	10.6771			0.227172			0.113586	+	0.993528i	$0.463766\pi$
$48$		0			0
$49$	65.0000			1.32653
$50$		0			0
$51$	−52.3068			−1.02562
$52$		0			0
$53$	−48.9898			−0.924336			−0.462168	−	0.886792i	$-0.652928\pi$
$53$	−48.9898			−0.924336			−0.462168	+	0.886792i	$0.652928\pi$
$54$		0			0
$55$	−21.3542	+	38.0000i	−0.388257	+	0.690909i
$56$		0			0
$57$		−	64.0625i		−	1.12390i
$58$		0			0
$59$	43.5890			0.738796			0.369398	−	0.929271i	$-0.379564\pi$
$59$	43.5890			0.738796			0.369398	+	0.929271i	$0.379564\pi$
$60$		0			0
$61$			26.1534i			0.428744i	0.976752	+	0.214372i	$0.0687705\pi$
$61$			26.1534i			0.428744i	−0.976752	+	0.214372i	$0.931229\pi$
$62$		0			0
$63$	−32.0312			−0.508432
$64$		0			0
$65$	−48.0000	+	85.4166i	−0.738462	+	1.31410i
$66$		0			0
$67$			7.34847i			0.109679i	0.998495	+	0.0548393i	$0.0174647\pi$
$67$			7.34847i			0.109679i	−0.998495	+	0.0548393i	$0.982535\pi$
$68$		0			0
$69$			26.1534i			0.379035i
$70$		0			0
$71$			84.0000i			1.18310i	0.806269	+	0.591549i	$0.201483\pi$
$71$			84.0000i			1.18310i	−0.806269	+	0.591549i	$0.798517\pi$
$72$		0			0
$73$			106.771i			1.46261i	0.682049	+	0.731307i	$0.261089\pi$
$73$			106.771i			1.46261i	−0.682049	+	0.731307i	$0.738911\pi$
$74$		0			0
$75$	52.3068	−	31.8434i	0.697424	−	0.424578i
$76$		0			0
$77$	93.0806			1.20884
$78$		0			0
$79$		−	100.000i		−	1.26582i	−0.774224	−	0.632911i	$-0.781860\pi$
$79$		−	100.000i		−	1.26582i	0.774224	−	0.632911i	$-0.218140\pi$
$80$		0			0
$81$	−45.0000			−0.555556
$82$		0			0
$83$			17.1464i			0.206583i	0.994651	+	0.103292i	$0.0329375\pi$
$83$			17.1464i			0.206583i	−0.994651	+	0.103292i	$0.967062\pi$
$84$		0			0
$85$	93.0806	+	52.3068i	1.09507	+	0.615374i
$86$		0			0
$87$	85.4166			0.981800
$88$		0			0
$89$	150.000			1.68539			0.842697	−	0.538389i	$-0.180967\pi$
$89$	150.000			1.68539			0.842697	+	0.538389i	$0.180967\pi$
$90$		0			0
$91$	209.227			2.29920
$92$		0			0
$93$	−9.79796			−0.105354
$94$		0			0
$95$	−64.0625	+	114.000i	−0.674342	+	1.20000i
$96$		0			0
$97$		−	21.3542i		−	0.220146i	−0.993924	−	0.110073i	$-0.964892\pi$
$97$		−	21.3542i		−	0.220146i	0.993924	−	0.110073i	$-0.0351085\pi$
$98$		0			0
$99$	−26.1534			−0.264176
$100$		0			0
$101$			69.7424i			0.690519i	0.938507	+	0.345259i	$0.112209\pi$
$101$			69.7424i			0.690519i	−0.938507	+	0.345259i	$0.887791\pi$
$102$		0			0
$103$	−32.0312			−0.310983			−0.155491	−	0.987837i	$-0.549696\pi$
$103$	−32.0312			−0.310983			−0.155491	+	0.987837i	$0.549696\pi$
$104$		0			0
$105$	−114.000	−	64.0625i	−1.08571	−	0.610119i
$106$		0			0
$107$		−	149.419i		−	1.39644i	−0.715884	−	0.698219i	$-0.753976\pi$
$107$		−	149.419i		−	1.39644i	0.715884	−	0.698219i	$-0.246024\pi$
$108$		0			0
$109$			183.074i			1.67958i	0.542915	+	0.839788i	$0.317321\pi$
$109$			183.074i			1.67958i	−0.542915	+	0.839788i	$0.682679\pi$
$110$		0			0
$111$		−	36.0000i		−	0.324324i
$112$		0			0
$113$		−	170.833i		−	1.51180i	−0.654688	−	0.755899i	$-0.727200\pi$
$113$		−	170.833i		−	1.51180i	0.654688	−	0.755899i	$-0.272800\pi$
$114$		0			0
$115$	26.1534	−	46.5403i	0.227421	−	0.404698i
$116$		0			0
$117$	−58.7878			−0.502459
$118$		0			0
$119$		−	228.000i		−	1.91597i
$120$		0			0
$121$	−45.0000			−0.371901
$122$		0			0
$123$		−	58.7878i		−	0.477949i
$124$		0			0
$125$	−124.924	+	4.35890i	−0.999392	+	0.0348712i
$126$		0			0
$127$	−53.3854			−0.420357			−0.210179	−	0.977663i	$-0.567405\pi$
$127$	−53.3854			−0.420357			−0.210179	+	0.977663i	$0.567405\pi$
$128$		0			0
$129$	138.000			1.06977
$130$		0			0
$131$	−95.8958			−0.732029			−0.366014	−	0.930609i	$-0.619278\pi$
$131$	−95.8958			−0.732029			−0.366014	+	0.930609i	$0.619278\pi$
$132$		0			0
$133$	279.242			2.09956
$134$		0			0
$135$	128.125	+	72.0000i	0.949074	+	0.533333i
$136$		0			0
$137$			128.125i			0.935219i	0.883935	+	0.467609i	$0.154885\pi$
$137$			128.125i			0.935219i	−0.883935	+	0.467609i	$0.845115\pi$
$138$		0			0
$139$	235.381			1.69339			0.846693	−	0.532082i	$-0.178590\pi$
$139$	235.381			1.69339			0.846693	+	0.532082i	$0.178590\pi$
$140$		0			0
$141$			26.1534i			0.185485i
$142$		0			0
$143$	170.833			1.19464
$144$		0			0
$145$	−152.000	−	85.4166i	−1.04828	−	0.589080i
$146$		0			0
$147$			159.217i			1.08311i
$148$		0			0
$149$		−	95.8958i		−	0.643596i	−0.946808	−	0.321798i	$-0.895713\pi$
$149$		−	95.8958i		−	0.643596i	0.946808	−	0.321798i	$-0.104287\pi$
$150$		0			0
$151$			160.000i			1.05960i	0.848122	+	0.529801i	$0.177734\pi$
$151$			160.000i			1.05960i	−0.848122	+	0.529801i	$0.822266\pi$
$152$		0			0
$153$			64.0625i			0.418709i
$154$		0			0
$155$	17.4356	+	9.79796i	0.112488	+	0.0632126i
$156$		0			0
$157$	34.2929			0.218426			0.109213	−	0.994018i	$-0.465167\pi$
$157$	34.2929			0.218426			0.109213	+	0.994018i	$0.465167\pi$
$158$		0			0
$159$		−	120.000i		−	0.754717i
$160$		0			0
$161$	−114.000			−0.708075
$162$		0			0
$163$			120.025i			0.736350i	0.929757	+	0.368175i	$0.120017\pi$
$163$			120.025i			0.736350i	−0.929757	+	0.368175i	$0.879983\pi$
$164$		0			0
$165$	−93.0806	−	52.3068i	−0.564125	−	0.317011i
$166$		0			0
$167$	−288.281			−1.72623			−0.863117	−	0.505004i	$-0.831491\pi$
$167$	−288.281			−1.72623			−0.863117	+	0.505004i	$0.831491\pi$
$168$		0			0
$169$	215.000			1.27219
$170$		0			0
$171$	−78.4602			−0.458831
$172$		0			0
$173$	24.4949			0.141589			0.0707945	−	0.997491i	$-0.477447\pi$
$173$	24.4949			0.141589			0.0707945	+	0.997491i	$0.477447\pi$
$174$		0			0
$175$	138.802	+	228.000i	0.793154	+	1.30286i
$176$		0			0
$177$			106.771i			0.603225i
$178$		0			0
$179$	−252.816			−1.41238			−0.706190	−	0.708022i	$-0.749588\pi$
$179$	−252.816			−1.41238			−0.706190	+	0.708022i	$0.749588\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−64.0625			−0.350068
$184$		0			0
$185$	−36.0000	+	64.0625i	−0.194595	+	0.346284i
$186$		0			0
$187$		−	186.161i		−	0.995515i
$188$		0			0
$189$		−	313.841i		−	1.66053i
$190$		0			0
$191$			24.0000i			0.125654i	0.998024	+	0.0628272i	$0.0200117\pi$
$191$			24.0000i			0.125654i	−0.998024	+	0.0628272i	$0.979988\pi$
$192$		0			0
$193$			106.771i			0.553216i	0.960983	+	0.276608i	$0.0892104\pi$
$193$			106.771i			0.553216i	−0.960983	+	0.276608i	$0.910790\pi$
$194$		0			0
$195$	−209.227	−	117.576i	−1.07296	−	0.602951i
$196$		0			0
$197$	−342.929			−1.74075			−0.870377	−	0.492386i	$-0.836125\pi$
$197$	−342.929			−1.74075			−0.870377	+	0.492386i	$0.836125\pi$
$198$		0			0
$199$		−	188.000i		−	0.944724i	−0.881405	−	0.472362i	$-0.843402\pi$
$199$		−	188.000i		−	0.944724i	0.881405	−	0.472362i	$-0.156598\pi$
$200$		0			0
$201$	−18.0000			−0.0895522
$202$		0			0
$203$			372.322i			1.83410i
$204$		0			0
$205$	−58.7878	+	104.614i	−0.286770	+	0.510310i
$206$		0			0
$207$	32.0312			0.154740
$208$		0			0
$209$	228.000			1.09091
$210$		0			0
$211$	−26.1534			−0.123950			−0.0619749	−	0.998078i	$-0.519740\pi$
$211$	−26.1534			−0.123950			−0.0619749	+	0.998078i	$0.519740\pi$
$212$		0			0
$213$	−205.757			−0.965996
$214$		0			0
$215$	−245.573	−	138.000i	−1.14220	−	0.641860i
$216$		0			0
$217$		−	42.7083i		−	0.196813i
$218$		0			0
$219$	−261.534			−1.19422
$220$		0			0
$221$		−	418.454i		−	1.89346i
$222$		0			0
$223$	−224.219			−1.00546			−0.502732	−	0.864442i	$-0.667672\pi$
$223$	−224.219			−1.00546			−0.502732	+	0.864442i	$0.667672\pi$
$224$		0			0
$225$	−39.0000	−	64.0625i	−0.173333	−	0.284722i
$226$		0			0
$227$			80.8332i			0.356093i	0.984022	+	0.178047i	$0.0569778\pi$
$227$			80.8332i			0.356093i	−0.984022	+	0.178047i	$0.943022\pi$
$228$		0			0
$229$			104.614i			0.456828i	0.973564	+	0.228414i	$0.0733539\pi$
$229$			104.614i			0.456828i	−0.973564	+	0.228414i	$0.926646\pi$
$230$		0			0
$231$			228.000i			0.987013i
$232$		0			0
$233$		−	320.312i		−	1.37473i	−0.726312	−	0.687366i	$-0.758767\pi$
$233$		−	320.312i		−	1.37473i	0.726312	−	0.687366i	$-0.241233\pi$
$234$		0			0
$235$	26.1534	−	46.5403i	0.111291	−	0.198044i
$236$		0			0
$237$	244.949			1.03354
$238$		0			0
$239$		−	108.000i		−	0.451883i	−0.974141	−	0.225941i	$-0.927454\pi$
$239$		−	108.000i		−	0.451883i	0.974141	−	0.225941i	$-0.0725458\pi$
$240$		0			0
$241$	−28.0000			−0.116183			−0.0580913	−	0.998311i	$-0.518501\pi$
$241$	−28.0000			−0.116183			−0.0580913	+	0.998311i	$0.518501\pi$
$242$		0			0
$243$			154.318i			0.635053i
$244$		0			0
$245$	159.217	−	283.328i	0.649865	−	1.15644i
$246$		0			0
$247$	512.500			2.07490
$248$		0			0
$249$	−42.0000			−0.168675
$250$		0			0
$251$	−252.816			−1.00724			−0.503618	−	0.863927i	$-0.667998\pi$
$251$	−252.816			−1.00724			−0.503618	+	0.863927i	$0.667998\pi$
$252$		0			0
$253$	−93.0806			−0.367908
$254$		0			0
$255$	−128.125	+	228.000i	−0.502451	+	0.894118i
$256$		0			0
$257$		−	298.958i		−	1.16326i	−0.813453	−	0.581631i	$-0.802415\pi$
$257$		−	298.958i		−	1.16326i	0.813453	−	0.581631i	$-0.197585\pi$
$258$		0			0
$259$	156.920			0.605870
$260$		0			0
$261$		−	104.614i		−	0.400818i
$262$		0			0
$263$	288.281			1.09613			0.548063	−	0.836437i	$-0.315365\pi$
$263$	288.281			1.09613			0.548063	+	0.836437i	$0.315365\pi$
$264$		0			0
$265$	−120.000	+	213.542i	−0.452830	+	0.805817i
$266$		0			0
$267$			367.423i			1.37612i
$268$		0			0
$269$			61.0246i			0.226857i	0.993546	+	0.113429i	$0.0361833\pi$
$269$			61.0246i			0.226857i	−0.993546	+	0.113429i	$0.963817\pi$
$270$		0			0
$271$		−	428.000i		−	1.57934i	−0.613535	−	0.789668i	$-0.710253\pi$
$271$		−	428.000i		−	1.57934i	0.613535	−	0.789668i	$-0.289747\pi$
$272$		0			0
$273$			512.500i			1.87729i
$274$		0			0
$275$	113.331	+	186.161i	0.412114	+	0.676950i
$276$		0			0
$277$	−73.4847			−0.265288			−0.132644	−	0.991164i	$-0.542347\pi$
$277$	−73.4847			−0.265288			−0.132644	+	0.991164i	$0.542347\pi$
$278$		0			0
$279$			12.0000i			0.0430108i
$280$		0			0
$281$	−204.000			−0.725979			−0.362989	−	0.931793i	$-0.618244\pi$
$281$	−204.000			−0.725979			−0.362989	+	0.931793i	$0.618244\pi$
$282$		0			0
$283$		−	404.166i		−	1.42815i	−0.700070	−	0.714074i	$-0.746848\pi$
$283$		−	404.166i		−	1.42815i	0.700070	−	0.714074i	$-0.253152\pi$
$284$		0			0
$285$	−279.242	−	156.920i	−0.979796	−	0.550598i
$286$		0			0
$287$	256.250			0.892857
$288$		0			0
$289$	−167.000			−0.577855
$290$		0			0
$291$	52.3068			0.179748
$292$		0			0
$293$	161.666			0.551762			0.275881	−	0.961192i	$-0.411030\pi$
$293$	161.666			0.551762			0.275881	+	0.961192i	$0.411030\pi$
$294$		0			0
$295$	106.771	−	190.000i	0.361935	−	0.644068i
$296$		0			0
$297$		−	256.250i		−	0.862794i
$298$		0			0
$299$	−209.227			−0.699756
$300$		0			0
$301$			601.528i			1.99843i
$302$		0			0
$303$	−170.833			−0.563806
$304$		0			0
$305$	114.000	+	64.0625i	0.373770	+	0.210041i
$306$		0			0
$307$		−	502.145i		−	1.63565i	−0.575465	−	0.817826i	$-0.695179\pi$
$307$		−	502.145i		−	1.63565i	0.575465	−	0.817826i	$-0.304821\pi$
$308$		0			0
$309$		−	78.4602i		−	0.253916i
$310$		0			0
$311$			408.000i			1.31190i	0.754806	+	0.655949i	$0.227731\pi$
$311$			408.000i			1.31190i	−0.754806	+	0.655949i	$0.772269\pi$
$312$		0			0
$313$		−	298.958i		−	0.955138i	−0.878594	−	0.477569i	$-0.841518\pi$
$313$		−	298.958i		−	0.955138i	0.878594	−	0.477569i	$-0.158482\pi$
$314$		0			0
$315$	−78.4602	+	139.621i	−0.249080	+	0.443241i
$316$		0			0
$317$	−186.161			−0.587259			−0.293630	−	0.955919i	$-0.594863\pi$
$317$	−186.161			−0.587259			−0.293630	+	0.955919i	$0.594863\pi$
$318$		0			0
$319$			304.000i			0.952978i
$320$		0			0
$321$	366.000			1.14019
$322$		0			0
$323$		−	558.484i		−	1.72905i
$324$		0			0
$325$	254.747	+	418.454i	0.783837	+	1.28755i
$326$		0			0
$327$	−448.437			−1.37137
$328$		0			0
$329$	−114.000			−0.346505
$330$		0			0
$331$	287.687			0.869146			0.434573	−	0.900637i	$-0.356899\pi$
$331$	287.687			0.869146			0.434573	+	0.900637i	$0.356899\pi$
$332$		0			0
$333$	−44.0908			−0.132405
$334$		0			0
$335$	32.0312	+	18.0000i	0.0956156	+	0.0537313i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	418.454			1.23438
$340$		0			0
$341$		−	34.8712i		−	0.102262i
$342$		0			0
$343$	−170.833			−0.498056
$344$		0			0
$345$	114.000	+	64.0625i	0.330435	+	0.185688i
$346$		0			0
$347$			413.964i			1.19298i	0.802621	+	0.596490i	$0.203438\pi$
$347$			413.964i			1.19298i	−0.802621	+	0.596490i	$0.796562\pi$
$348$		0			0
$349$			209.227i			0.599505i	0.954017	+	0.299752i	$0.0969041\pi$
$349$			209.227i			0.599505i	−0.954017	+	0.299752i	$0.903096\pi$
$350$		0			0
$351$		−	576.000i		−	1.64103i
$352$		0			0
$353$			213.542i			0.604934i	0.953160	+	0.302467i	$0.0978101\pi$
$353$			213.542i			0.604934i	−0.953160	+	0.302467i	$0.902190\pi$
$354$		0			0
$355$	366.148	+	205.757i	1.03140	+	0.579598i
$356$		0			0
$357$	558.484			1.56438
$358$		0			0
$359$			72.0000i			0.200557i	0.994959	+	0.100279i	$0.0319734\pi$
$359$			72.0000i			0.200557i	−0.994959	+	0.100279i	$0.968027\pi$
$360$		0			0
$361$	323.000			0.894737
$362$		0			0
$363$		−	110.227i		−	0.303656i
$364$		0			0
$365$	465.403	+	261.534i	1.27508	+	0.716531i
$366$		0			0
$367$	−523.177			−1.42555			−0.712775	−	0.701393i	$-0.752562\pi$
$367$	−523.177			−1.42555			−0.712775	+	0.701393i	$0.752562\pi$
$368$		0			0
$369$	−72.0000			−0.195122
$370$		0			0
$371$	523.068			1.40989
$372$		0			0
$373$	−93.0806			−0.249546			−0.124773	−	0.992185i	$-0.539820\pi$
$373$	−93.0806			−0.249546			−0.124773	+	0.992185i	$0.539820\pi$
$374$		0			0
$375$	−10.6771	−	306.000i	−0.0284722	−	0.816000i
$376$		0			0
$377$			683.333i			1.81255i
$378$		0			0
$379$	339.994			0.897082			0.448541	−	0.893762i	$-0.351944\pi$
$379$	339.994			0.897082			0.448541	+	0.893762i	$0.351944\pi$
$380$		0			0
$381$		−	130.767i		−	0.343220i
$382$		0			0
$383$	245.573			0.641182			0.320591	−	0.947218i	$-0.396118\pi$
$383$	245.573			0.641182			0.320591	+	0.947218i	$0.396118\pi$
$384$		0			0
$385$	228.000	−	405.729i	0.592208	−	1.05384i
$386$		0			0
$387$		−	169.015i		−	0.436731i
$388$		0			0
$389$			148.203i			0.380983i	0.981689	+	0.190492i	$0.0610082\pi$
$389$			148.203i			0.380983i	−0.981689	+	0.190492i	$0.938992\pi$
$390$		0			0
$391$			228.000i			0.583120i
$392$		0			0
$393$		−	234.896i		−	0.597699i
$394$		0			0
$395$	−435.890	−	244.949i	−1.10352	−	0.620124i
$396$		0			0
$397$	460.504			1.15996			0.579980	−	0.814631i	$-0.303060\pi$
$397$	460.504			1.15996			0.579980	+	0.814631i	$0.303060\pi$
$398$		0			0
$399$			684.000i			1.71429i
$400$		0			0
$401$	−330.000			−0.822943			−0.411471	−	0.911423i	$-0.634985\pi$
$401$	−330.000			−0.822943			−0.411471	+	0.911423i	$0.634985\pi$
$402$		0			0
$403$		−	78.3837i		−	0.194500i
$404$		0			0
$405$	−110.227	+	196.150i	−0.272166	+	0.484322i
$406$		0			0
$407$	128.125			0.314803
$408$		0			0
$409$	−536.000			−1.31051			−0.655257	−	0.755406i	$-0.727440\pi$
$409$	−536.000			−1.31051			−0.655257	+	0.755406i	$0.727440\pi$
$410$		0			0
$411$	−313.841			−0.763603
$412$		0			0
$413$	−465.403			−1.12688
$414$		0			0
$415$	74.7395	+	42.0000i	0.180095	+	0.101205i
$416$		0			0
$417$			576.562i			1.38264i
$418$		0			0
$419$	479.479			1.14434			0.572171	−	0.820135i	$-0.306102\pi$
$419$	479.479			1.14434			0.572171	+	0.820135i	$0.306102\pi$
$420$		0			0
$421$			496.914i			1.18032i	0.807287	+	0.590160i	$0.200935\pi$
$421$			496.914i			1.18032i	−0.807287	+	0.590160i	$0.799065\pi$
$422$		0			0
$423$	32.0312			0.0757240
$424$		0			0
$425$	456.000	−	277.604i	1.07294	−	0.653186i
$426$		0			0
$427$		−	279.242i		−	0.653962i
$428$		0			0
$429$			418.454i			0.975418i
$430$		0			0
$431$		−	180.000i		−	0.417633i	−0.977955	−	0.208817i	$-0.933039\pi$
$431$		−	180.000i		−	0.417633i	0.977955	−	0.208817i	$-0.0669612\pi$
$432$		0			0
$433$		−	363.021i		−	0.838385i	−0.907897	−	0.419192i	$-0.862313\pi$
$433$		−	363.021i		−	0.838385i	0.907897	−	0.419192i	$-0.137687\pi$
$434$		0			0
$435$	209.227	−	372.322i	0.480982	−	0.855914i
$436$		0			0
$437$	−279.242			−0.638997
$438$		0			0
$439$			136.000i			0.309795i	0.987931	+	0.154897i	$0.0495047\pi$
$439$			136.000i			0.309795i	−0.987931	+	0.154897i	$0.950495\pi$
$440$		0			0
$441$	195.000			0.442177
$442$		0			0
$443$			183.712i			0.414699i	0.978267	+	0.207350i	$0.0664838\pi$
$443$			183.712i			0.414699i	−0.978267	+	0.207350i	$0.933516\pi$
$444$		0			0
$445$	367.423	−	653.835i	0.825671	−	1.46929i
$446$		0			0
$447$	234.896			0.525494
$448$		0			0
$449$	−684.000			−1.52339			−0.761693	−	0.647939i	$-0.775631\pi$
$449$	−684.000			−1.52339			−0.761693	+	0.647939i	$0.775631\pi$
$450$		0			0
$451$	209.227			0.463918
$452$		0			0
$453$	−391.918			−0.865162
$454$		0			0
$455$	512.500	−	912.000i	1.12637	−	2.00440i
$456$		0			0
$457$		−	85.4166i		−	0.186907i	−0.995624	−	0.0934536i	$-0.970209\pi$
$457$		−	85.4166i		−	0.186907i	0.995624	−	0.0934536i	$-0.0297907\pi$
$458$		0			0
$459$	−627.681			−1.36750
$460$		0			0
$461$			348.712i			0.756425i	0.925719	+	0.378212i	$0.123461\pi$
$461$			348.712i			0.756425i	−0.925719	+	0.378212i	$0.876539\pi$
$462$		0			0
$463$	565.885			1.22221			0.611107	−	0.791548i	$-0.290724\pi$
$463$	565.885			1.22221			0.611107	+	0.791548i	$0.290724\pi$
$464$		0			0
$465$	−24.0000	+	42.7083i	−0.0516129	+	0.0918458i
$466$		0			0
$467$		−	105.328i		−	0.225542i	−0.993621	−	0.112771i	$-0.964027\pi$
$467$		−	105.328i		−	0.225542i	0.993621	−	0.112771i	$-0.0359726\pi$
$468$		0			0
$469$		−	78.4602i		−	0.167292i
$470$		0			0
$471$			84.0000i			0.178344i
$472$		0			0
$473$			491.146i			1.03836i
$474$		0			0
$475$	339.994	+	558.484i	0.715777	+	1.17576i
$476$		0			0
$477$	−146.969			−0.308112
$478$		0			0
$479$			552.000i			1.15240i	0.817308	+	0.576200i	$0.195465\pi$
$479$			552.000i			1.15240i	−0.817308	+	0.576200i	$0.804535\pi$
$480$		0			0
$481$	288.000			0.598753
$482$		0			0
$483$		−	279.242i		−	0.578140i
$484$		0			0
$485$	−93.0806	−	52.3068i	−0.191919	−	0.107849i
$486$		0			0
$487$	−224.219			−0.460408			−0.230204	−	0.973142i	$-0.573939\pi$
$487$	−224.219			−0.460408			−0.230204	+	0.973142i	$0.573939\pi$
$488$		0			0
$489$	−294.000			−0.601227
$490$		0			0
$491$	−479.479			−0.976535			−0.488268	−	0.872694i	$-0.662371\pi$
$491$	−479.479			−0.976535			−0.488268	+	0.872694i	$0.662371\pi$
$492$		0			0
$493$	744.645			1.51044
$494$		0			0
$495$	−64.0625	+	114.000i	−0.129419	+	0.230303i
$496$		0			0
$497$		−	896.875i		−	1.80458i
$498$		0			0
$499$	−287.687			−0.576528			−0.288264	−	0.957551i	$-0.593078\pi$
$499$	−287.687			−0.576528			−0.288264	+	0.957551i	$0.593078\pi$
$500$		0			0
$501$		−	706.142i		−	1.40946i
$502$		0			0
$503$	−715.364			−1.42220			−0.711098	−	0.703093i	$-0.751802\pi$
$503$	−715.364			−1.42220			−0.711098	+	0.703093i	$0.751802\pi$
$504$		0			0
$505$	304.000	+	170.833i	0.601980	+	0.338284i
$506$		0			0
$507$			526.640i			1.03874i
$508$		0			0
$509$		−	802.037i		−	1.57571i	−0.615860	−	0.787856i	$-0.711191\pi$
$509$		−	802.037i		−	1.57571i	0.615860	−	0.787856i	$-0.288809\pi$
$510$		0			0
$511$		−	1140.00i		−	2.23092i
$512$		0			0
$513$		−	768.750i		−	1.49854i
$514$		0			0
$515$	−78.4602	+	139.621i	−0.152350	+	0.271109i
$516$		0			0
$517$	−93.0806			−0.180040
$518$		0			0
$519$			60.0000i			0.115607i
$520$		0			0
$521$	−102.000			−0.195777			−0.0978887	−	0.995197i	$-0.531209\pi$
$521$	−102.000			−0.195777			−0.0978887	+	0.995197i	$0.531209\pi$
$522$		0			0
$523$			575.630i			1.10063i	0.834957	+	0.550316i	$0.185493\pi$
$523$			575.630i			1.10063i	−0.834957	+	0.550316i	$0.814507\pi$
$524$		0			0
$525$	−558.484	+	339.994i	−1.06378	+	0.647608i
$526$		0			0
$527$	−85.4166			−0.162081
$528$		0			0
$529$	−415.000			−0.784499
$530$		0			0
$531$	130.767			0.246265
$532$		0			0
$533$	470.302			0.882368
$534$		0			0
$535$	−651.302	−	366.000i	−1.21739	−	0.684112i
$536$		0			0
$537$		−	619.271i		−	1.15320i
$538$		0			0
$539$	−566.657			−1.05131
$540$		0			0
$541$		−	836.909i		−	1.54697i	−0.633817	−	0.773483i	$-0.718513\pi$
$541$		−	836.909i		−	1.54697i	0.633817	−	0.773483i	$-0.281487\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	798.000	+	448.437i	1.46422	+	0.822821i
$546$		0			0
$547$			560.933i			1.02547i	0.858546	+	0.512736i	$0.171368\pi$
$547$			560.933i			1.02547i	−0.858546	+	0.512736i	$0.828632\pi$
$548$		0			0
$549$			78.4602i			0.142915i
$550$		0			0
$551$			912.000i			1.65517i
$552$		0			0
$553$			1067.71i			1.93076i
$554$		0			0
$555$	−156.920	−	88.1816i	−0.282739	−	0.158886i
$556$		0			0
$557$	−788.736			−1.41604			−0.708021	−	0.706191i	$-0.750412\pi$
$557$	−788.736			−1.41604			−0.708021	+	0.706191i	$0.750412\pi$
$558$		0			0
$559$			1104.00i			1.97496i
$560$		0			0
$561$	456.000			0.812834
$562$		0			0
$563$			149.419i			0.265398i	0.991156	+	0.132699i	$0.0423643\pi$
$563$			149.419i			0.265398i	−0.991156	+	0.132699i	$0.957636\pi$
$564$		0			0
$565$	−744.645	−	418.454i	−1.31796	−	0.740627i
$566$		0			0
$567$	480.469			0.847387
$568$		0			0
$569$	−144.000			−0.253076			−0.126538	−	0.991962i	$-0.540386\pi$
$569$	−144.000			−0.253076			−0.126538	+	0.991962i	$0.540386\pi$
$570$		0			0
$571$	−496.914			−0.870253			−0.435127	−	0.900369i	$-0.643296\pi$
$571$	−496.914			−0.870253			−0.435127	+	0.900369i	$0.643296\pi$
$572$		0			0
$573$	−58.7878			−0.102596
$574$		0			0
$575$	−138.802	−	228.000i	−0.241395	−	0.396522i
$576$		0			0
$577$			384.375i			0.666161i	0.942898	+	0.333080i	$0.108088\pi$
$577$			384.375i			0.666161i	−0.942898	+	0.333080i	$0.891912\pi$
$578$		0			0
$579$	−261.534			−0.451699
$580$		0			0
$581$		−	183.074i		−	0.315101i
$582$		0			0
$583$	427.083			0.732561
$584$		0			0
$585$	−144.000	+	256.250i	−0.246154	+	0.438034i
$586$		0			0
$587$		−	869.569i		−	1.48138i	−0.671848	−	0.740689i	$-0.734499\pi$
$587$		−	869.569i		−	1.48138i	0.671848	−	0.740689i	$-0.265501\pi$
$588$		0			0
$589$		−	104.614i		−	0.177612i
$590$		0			0
$591$		−	840.000i		−	1.42132i
$592$		0			0
$593$			213.542i			0.360104i	0.983657	+	0.180052i	$0.0576266\pi$
$593$			213.542i			0.360104i	−0.983657	+	0.180052i	$0.942373\pi$
$594$		0			0
$595$	−993.829	−	558.484i	−1.67030	−	0.938628i
$596$		0			0
$597$	460.504			0.771364
$598$		0			0
$599$			300.000i			0.500835i	0.968138	+	0.250417i	$0.0805678\pi$
$599$			300.000i			0.500835i	−0.968138	+	0.250417i	$0.919432\pi$
$600$		0			0
$601$	740.000			1.23128			0.615641	−	0.788027i	$-0.288897\pi$
$601$	740.000			1.23128			0.615641	+	0.788027i	$0.288897\pi$
$602$		0			0
$603$			22.0454i			0.0365595i
$604$		0			0
$605$	−110.227	+	196.150i	−0.182193	+	0.324216i
$606$		0			0
$607$	608.593			1.00263			0.501313	−	0.865266i	$-0.332851\pi$
$607$	608.593			1.00263			0.501313	+	0.865266i	$0.332851\pi$
$608$		0			0
$609$	−912.000			−1.49754
$610$		0			0
$611$	−209.227			−0.342434
$612$		0			0
$613$	−431.110			−0.703279			−0.351640	−	0.936135i	$-0.614376\pi$
$613$	−431.110			−0.703279			−0.351640	+	0.936135i	$0.614376\pi$
$614$		0			0
$615$	−256.250	−	144.000i	−0.416666	−	0.234146i
$616$		0			0
$617$			106.771i			0.173048i	0.996250	+	0.0865241i	$0.0275760\pi$
$617$			106.771i			0.173048i	−0.996250	+	0.0865241i	$0.972424\pi$
$618$		0			0
$619$	183.074			0.295757			0.147879	−	0.989006i	$-0.452755\pi$
$619$	183.074			0.295757			0.147879	+	0.989006i	$0.452755\pi$
$620$		0			0
$621$			313.841i			0.505380i
$622$		0			0
$623$	−1601.56			−2.57073
$624$		0			0
$625$	−287.000	+	555.208i	−0.459200	+	0.888333i
$626$		0			0
$627$			558.484i			0.890724i
$628$		0			0
$629$		−	313.841i		−	0.498952i
$630$		0			0
$631$			320.000i			0.507132i	0.967318	+	0.253566i	$0.0816034\pi$
$631$			320.000i			0.507132i	−0.967318	+	0.253566i	$0.918397\pi$
$632$		0			0
$633$		−	64.0625i		−	0.101205i
$634$		0			0
$635$	−130.767	+	232.702i	−0.205932	+	0.366459i
$636$		0			0
$637$	−1273.73			−1.99958
$638$		0			0
$639$			252.000i			0.394366i
$640$		0			0
$641$	−420.000			−0.655226			−0.327613	−	0.944812i	$-0.606244\pi$
$641$	−420.000			−0.655226			−0.327613	+	0.944812i	$0.606244\pi$
$642$		0			0
$643$			1045.93i			1.62664i	0.581814	+	0.813322i	$0.302343\pi$
$643$			1045.93i			1.62664i	−0.581814	+	0.813322i	$0.697657\pi$
$644$		0			0
$645$	338.030	−	601.528i	0.524077	−	0.932602i
$646$		0			0
$647$	544.531			0.841624			0.420812	−	0.907148i	$-0.361745\pi$
$647$	544.531			0.841624			0.420812	+	0.907148i	$0.361745\pi$
$648$		0			0
$649$	−380.000			−0.585516
$650$		0			0
$651$	104.614			0.160697
$652$		0			0
$653$	509.494			0.780236			0.390118	−	0.920765i	$-0.372434\pi$
$653$	509.494			0.780236			0.390118	+	0.920765i	$0.372434\pi$
$654$		0			0
$655$	−234.896	+	418.000i	−0.358619	+	0.638168i
$656$		0			0
$657$			320.312i			0.487538i
$658$		0			0
$659$	514.350			0.780501			0.390250	−	0.920709i	$-0.372388\pi$
$659$	514.350			0.780501			0.390250	+	0.920709i	$0.372388\pi$
$660$		0			0
$661$			26.1534i			0.0395664i	0.999804	+	0.0197832i	$0.00629760\pi$
$661$			26.1534i			0.0395664i	−0.999804	+	0.0197832i	$0.993702\pi$
$662$		0			0
$663$	1025.00			1.54600
$664$		0			0
$665$	684.000	−	1217.19i	1.02857	−	1.83036i
$666$		0			0
$667$		−	372.322i		−	0.558205i
$668$		0			0
$669$		−	549.221i		−	0.820959i
$670$		0			0
$671$		−	228.000i		−	0.339791i
$672$		0			0
$673$		−	533.854i		−	0.793245i	−0.917982	−	0.396623i	$-0.870182\pi$
$673$		−	533.854i		−	0.793245i	0.917982	−	0.396623i	$-0.129818\pi$
$674$		0			0
$675$	627.681	−	382.120i	0.929898	−	0.566104i
$676$		0			0
$677$	1116.97			1.64988			0.824939	−	0.565222i	$-0.191209\pi$
$677$	1116.97			1.64988			0.824939	+	0.565222i	$0.191209\pi$
$678$		0			0
$679$			228.000i			0.335788i
$680$		0			0
$681$	−198.000			−0.290749
$682$		0			0
$683$			1256.59i			1.83981i	0.392145	+	0.919904i	$0.371733\pi$
$683$			1256.59i			1.83981i	−0.392145	+	0.919904i	$0.628267\pi$
$684$		0			0
$685$	558.484	+	313.841i	0.815305	+	0.458162i
$686$		0			0
$687$	−256.250			−0.372998
$688$		0			0
$689$	960.000			1.39332
$690$		0			0
$691$	−1072.29			−1.55179			−0.775897	−	0.630860i	$-0.782702\pi$
$691$	−1072.29			−1.55179			−0.775897	+	0.630860i	$0.782702\pi$
$692$		0			0
$693$	279.242			0.402946
$694$		0			0
$695$	576.562	−	1026.00i	0.829586	−	1.47626i
$696$		0			0
$697$		−	512.500i		−	0.735294i
$698$		0			0
$699$	784.602			1.12246
$700$		0			0
$701$			43.5890i			0.0621812i	0.999517	+	0.0310906i	$0.00989803\pi$
$701$			43.5890i			0.0621812i	−0.999517	+	0.0310906i	$0.990102\pi$
$702$		0			0
$703$	384.375			0.546764
$704$		0			0
$705$	114.000	+	64.0625i	0.161702	+	0.0908688i
$706$		0			0
$707$		−	744.645i		−	1.05325i
$708$		0			0
$709$		−	209.227i		−	0.295102i	−0.989054	−	0.147551i	$-0.952861\pi$
$709$		−	209.227i		−	0.295102i	0.989054	−	0.147551i	$-0.0471390\pi$
$710$		0			0
$711$		−	300.000i		−	0.421941i
$712$		0			0
$713$			42.7083i			0.0598995i
$714$		0			0
$715$	418.454	−	744.645i	0.585251	−	1.04146i
$716$		0			0
$717$	264.545			0.368961
$718$		0			0
$719$		−	612.000i		−	0.851182i	−0.904916	−	0.425591i	$-0.860066\pi$
$719$		−	612.000i		−	0.851182i	0.904916	−	0.425591i	$-0.139934\pi$
$720$		0			0
$721$	342.000			0.474341
$722$		0			0
$723$		−	68.5857i		−	0.0948627i
$724$		0			0
$725$	−744.645	+	453.325i	−1.02710	+	0.625277i
$726$		0			0
$727$	−1334.63			−1.83581			−0.917906	−	0.396799i	$-0.870121\pi$
$727$	−1334.63			−1.83581			−0.917906	+	0.396799i	$0.870121\pi$
$728$		0			0
$729$	−783.000			−1.07407
$730$		0			0
$731$	1203.06			1.64577
$732$		0			0
$733$	−220.454			−0.300756			−0.150378	−	0.988629i	$-0.548049\pi$
$733$	−220.454			−0.300756			−0.150378	+	0.988629i	$0.548049\pi$
$734$		0			0
$735$	694.010	+	390.000i	0.944231	+	0.530612i
$736$		0			0
$737$		−	64.0625i		−	0.0869233i
$738$		0			0
$739$	−1438.44			−1.94646			−0.973232	−	0.229826i	$-0.926184\pi$
$739$	−1438.44			−1.94646			−0.973232	+	0.229826i	$0.926184\pi$
$740$		0			0
$741$			1255.36i			1.69415i
$742$		0			0
$743$	−138.802			−0.186813			−0.0934065	−	0.995628i	$-0.529776\pi$
$743$	−138.802			−0.186813			−0.0934065	+	0.995628i	$0.529776\pi$
$744$		0			0
$745$	−418.000	−	234.896i	−0.561074	−	0.315296i
$746$		0			0
$747$			51.4393i			0.0688612i
$748$		0			0
$749$			1595.36i			2.12998i
$750$		0			0
$751$		−	608.000i		−	0.809587i	−0.914408	−	0.404794i	$-0.867343\pi$
$751$		−	608.000i		−	0.809587i	0.914408	−	0.404794i	$-0.132657\pi$
$752$		0			0
$753$		−	619.271i		−	0.822404i
$754$		0			0
$755$	697.424	+	391.918i	0.923740	+	0.519097i
$756$		0			0
$757$	−592.777			−0.783060			−0.391530	−	0.920165i	$-0.628054\pi$
$757$	−592.777			−0.783060			−0.391530	+	0.920165i	$0.628054\pi$
$758$		0			0
$759$		−	228.000i		−	0.300395i
$760$		0			0
$761$	−642.000			−0.843627			−0.421813	−	0.906683i	$-0.638606\pi$
$761$	−642.000			−0.843627			−0.421813	+	0.906683i	$0.638606\pi$
$762$		0			0
$763$		−	1954.69i		−	2.56185i
$764$		0			0
$765$	279.242	+	156.920i	0.365022	+	0.205125i
$766$		0			0
$767$	−854.166			−1.11365
$768$		0			0
$769$	−230.000			−0.299090			−0.149545	−	0.988755i	$-0.547781\pi$
$769$	−230.000			−0.299090			−0.149545	+	0.988755i	$0.547781\pi$
$770$		0			0
$771$	732.295			0.949799
$772$		0			0
$773$	−88.1816			−0.114077			−0.0570386	−	0.998372i	$-0.518166\pi$
$773$	−88.1816			−0.114077			−0.0570386	+	0.998372i	$0.518166\pi$
$774$		0			0
$775$	85.4166	−	52.0000i	0.110215	−	0.0670968i
$776$		0			0
$777$			384.375i			0.494691i
$778$		0			0
$779$	627.681			0.805753
$780$		0			0
$781$		−	732.295i		−	0.937638i
$782$		0			0
$783$	1025.00			1.30907
$784$		0			0
$785$	84.0000	−	149.479i	0.107006	−	0.190419i
$786$		0			0
$787$			232.702i			0.295682i	0.989011	+	0.147841i	$0.0472323\pi$
$787$			232.702i			0.295682i	−0.989011	+	0.147841i	$0.952768\pi$
$788$		0			0
$789$			706.142i			0.894983i
$790$		0			0
$791$			1824.00i			2.30594i
$792$		0			0
$793$		−	512.500i		−	0.646280i
$794$		0			0
$795$	−523.068	−	293.939i	−0.657947	−	0.369734i
$796$		0			0
$797$	989.594			1.24165			0.620824	−	0.783950i	$-0.286798\pi$
$797$	989.594			1.24165			0.620824	+	0.783950i	$0.286798\pi$
$798$		0			0
$799$			228.000i			0.285357i
$800$		0			0
$801$	450.000			0.561798
$802$		0			0
$803$		−	930.806i		−	1.15916i
$804$		0			0
$805$	−279.242	+	496.914i	−0.346884	+	0.617285i
$806$		0			0
$807$	−149.479			−0.185228
$808$		0			0
$809$	906.000			1.11990			0.559951	−	0.828526i	$-0.310820\pi$
$809$	906.000			1.11990			0.559951	+	0.828526i	$0.310820\pi$
$810$		0			0
$811$	1281.52			1.58017			0.790084	−	0.612999i	$-0.210037\pi$
$811$	1281.52			1.58017			0.790084	+	0.612999i	$0.210037\pi$
$812$		0			0
$813$	1048.38			1.28952
$814$		0			0
$815$	523.177	+	294.000i	0.641935	+	0.360736i
$816$		0			0
$817$			1473.44i			1.80347i
$818$		0			0
$819$	627.681			0.766400
$820$		0			0
$821$			409.737i			0.499070i	0.968366	+	0.249535i	$0.0802778\pi$
$821$			409.737i			0.499070i	−0.968366	+	0.249535i	$0.919722\pi$
$822$		0			0
$823$	−373.698			−0.454068			−0.227034	−	0.973887i	$-0.572903\pi$
$823$	−373.698			−0.454068			−0.227034	+	0.973887i	$0.572903\pi$
$824$		0			0
$825$	−456.000	+	277.604i	−0.552727	+	0.336490i
$826$		0			0
$827$			169.015i			0.204371i	0.994765	+	0.102185i	$0.0325835\pi$
$827$			169.015i			0.204371i	−0.994765	+	0.102185i	$0.967416\pi$
$828$		0			0
$829$			130.767i			0.157741i	0.996885	+	0.0788703i	$0.0251313\pi$
$829$			130.767i			0.157741i	−0.996885	+	0.0788703i	$0.974869\pi$
$830$		0			0
$831$		−	180.000i		−	0.216606i
$832$		0			0
$833$			1388.02i			1.66629i
$834$		0			0
$835$	−706.142	+	1256.59i	−0.845679	+	1.50490i
$836$		0			0
$837$	−117.576			−0.140473
$838$		0			0
$839$		−	816.000i		−	0.972586i	−0.873796	−	0.486293i	$-0.838349\pi$
$839$		−	816.000i		−	0.972586i	0.873796	−	0.486293i	$-0.161651\pi$
$840$		0			0
$841$	−375.000			−0.445898
$842$		0			0
$843$		−	499.696i		−	0.592759i
$844$		0			0
$845$	526.640	−	937.163i	0.623243	−	1.10907i
$846$		0			0
$847$	480.469			0.567259
$848$		0			0
$849$	990.000			1.16608
$850$		0			0
$851$	−156.920			−0.184395
$852$		0			0
$853$	450.706			0.528378			0.264189	−	0.964471i	$-0.414896\pi$
$853$	450.706			0.528378			0.264189	+	0.964471i	$0.414896\pi$
$854$		0			0
$855$	−192.187	+	342.000i	−0.224781	+	0.400000i
$856$		0			0
$857$			512.500i			0.598016i	0.954251	+	0.299008i	$0.0966557\pi$
$857$			512.500i			0.598016i	−0.954251	+	0.299008i	$0.903344\pi$
$858$		0			0
$859$	496.914			0.578480			0.289240	−	0.957257i	$-0.406597\pi$
$859$	496.914			0.578480			0.289240	+	0.957257i	$0.406597\pi$
$860$		0			0
$861$			627.681i			0.729014i
$862$		0			0
$863$	53.3854			0.0618602			0.0309301	−	0.999522i	$-0.490153\pi$
$863$	53.3854			0.0618602			0.0309301	+	0.999522i	$0.490153\pi$
$864$		0			0
$865$	60.0000	−	106.771i	0.0693642	−	0.123434i
$866$		0			0
$867$		−	409.065i		−	0.471816i
$868$		0			0
$869$			871.780i			1.00320i
$870$		0			0
$871$		−	144.000i		−	0.165327i
$872$		0			0
$873$		−	64.0625i		−	0.0733820i
$874$		0			0
$875$	1333.82	−	46.5403i	1.52437	−	0.0531889i
$876$		0			0
$877$	−230.252			−0.262545			−0.131273	−	0.991346i	$-0.541906\pi$
$877$	−230.252			−0.262545			−0.131273	+	0.991346i	$0.541906\pi$
$878$		0			0
$879$			396.000i			0.450512i
$880$		0			0
$881$	−1320.00			−1.49830			−0.749149	−	0.662402i	$-0.769537\pi$
$881$	−1320.00			−1.49830			−0.749149	+	0.662402i	$0.769537\pi$
$882$		0			0
$883$			1050.83i			1.19007i	0.803700	+	0.595035i	$0.202862\pi$
$883$			1050.83i			1.19007i	−0.803700	+	0.595035i	$0.797138\pi$
$884$		0			0
$885$	465.403	+	261.534i	0.525879	+	0.295519i
$886$		0			0
$887$	117.448			0.132410			0.0662051	−	0.997806i	$-0.478911\pi$
$887$	117.448			0.132410			0.0662051	+	0.997806i	$0.478911\pi$
$888$		0			0
$889$	570.000			0.641170
$890$		0			0
$891$	392.301			0.440293
$892$		0			0
$893$	−279.242			−0.312701
$894$		0			0
$895$	−619.271	+	1102.00i	−0.691922	+	1.23128i
$896$		0			0
$897$		−	512.500i		−	0.571349i
$898$		0			0
$899$	139.485			0.155155
$900$		0			0
$901$		−	1046.14i		−	1.16108i
$902$		0			0
$903$	−1473.44			−1.63171
$904$		0			0
$905$		0			0
$906$		0			0
$907$			585.428i			0.645455i	0.946492	+	0.322728i	$0.104600\pi$
$907$			585.428i			0.645455i	−0.946492	+	0.322728i	$0.895400\pi$
$908$		0			0
$909$			209.227i			0.230173i
$910$		0			0
$911$			564.000i			0.619100i	0.950883	+	0.309550i	$0.100178\pi$
$911$			564.000i			0.619100i	−0.950883	+	0.309550i	$0.899822\pi$
$912$		0			0
$913$		−	149.479i		−	0.163723i
$914$		0			0
$915$	−156.920	+	279.242i	−0.171498	+	0.305182i
$916$		0			0
$917$	1023.89			1.11656
$918$		0			0
$919$		−	1124.00i		−	1.22307i	−0.791218	−	0.611534i	$-0.790553\pi$
$919$		−	1124.00i		−	1.22307i	0.791218	−	0.611534i	$-0.209447\pi$
$920$		0			0
$921$	1230.00			1.33550
$922$		0			0
$923$		−	1646.06i		−	1.78338i
$924$		0			0
$925$	191.060	+	313.841i	0.206552	+	0.339287i
$926$		0			0
$927$	−96.0937			−0.103661
$928$		0			0
$929$	−276.000			−0.297094			−0.148547	−	0.988905i	$-0.547460\pi$
$929$	−276.000			−0.297094			−0.148547	+	0.988905i	$0.547460\pi$
$930$		0			0
$931$	−1699.97			−1.82596
$932$		0			0
$933$	−999.392			−1.07116
$934$		0			0
$935$	−811.458	−	456.000i	−0.867869	−	0.487701i
$936$		0			0
$937$		−	619.271i		−	0.660908i	−0.943822	−	0.330454i	$-0.892798\pi$
$937$		−	619.271i		−	0.660908i	0.943822	−	0.330454i	$-0.107202\pi$
$938$		0			0
$939$	732.295			0.779867
$940$		0			0
$941$			1115.88i			1.18584i	0.805260	+	0.592921i	$0.202026\pi$
$941$			1115.88i			1.18584i	−0.805260	+	0.592921i	$0.797974\pi$
$942$		0			0
$943$	−256.250			−0.271739
$944$		0			0
$945$	−1368.00	−	768.750i	−1.44762	−	0.813492i
$946$		0			0
$947$		−	364.974i		−	0.385400i	−0.981258	−	0.192700i	$-0.938276\pi$
$947$		−	364.974i		−	0.385400i	0.981258	−	0.192700i	$-0.0617244\pi$
$948$		0			0
$949$		−	2092.27i		−	2.20471i
$950$		0			0
$951$		−	456.000i		−	0.479495i
$952$		0			0
$953$		−	341.667i		−	0.358517i	−0.983802	−	0.179258i	$-0.942630\pi$
$953$		−	341.667i		−	0.358517i	0.983802	−	0.179258i	$-0.0573698\pi$
$954$		0			0
$955$	104.614	+	58.7878i	0.109543	+	0.0615579i
$956$		0			0
$957$	−744.645			−0.778103
$958$		0			0
$959$		−	1368.00i		−	1.42649i
$960$		0			0
$961$	945.000			0.983351
$962$		0			0
$963$		−	448.257i		−	0.465479i
$964$		0			0
$965$	465.403	+	261.534i	0.482283	+	0.271020i
$966$		0			0
$967$	1548.18			1.60101			0.800505	−	0.599326i	$-0.204565\pi$
$967$	1548.18			1.60101			0.800505	+	0.599326i	$0.204565\pi$
$968$		0			0
$969$	1368.00			1.41176
$970$		0			0
$971$	−828.191			−0.852926			−0.426463	−	0.904505i	$-0.640241\pi$
$971$	−828.191			−0.852926			−0.426463	+	0.904505i	$0.640241\pi$
$972$		0			0
$973$	−2513.18			−2.58292
$974$		0			0
$975$	−1025.00	+	624.000i	−1.05128	+	0.640000i
$976$		0			0
$977$		−	1558.85i		−	1.59555i	−0.602955	−	0.797776i	$-0.706010\pi$
$977$		−	1558.85i		−	1.59555i	0.602955	−	0.797776i	$-0.293990\pi$
$978$		0			0
$979$	−1307.67			−1.33572
$980$		0			0
$981$			549.221i			0.559859i
$982$		0			0
$983$	843.489			0.858076			0.429038	−	0.903286i	$-0.358853\pi$
$983$	843.489			0.858076			0.429038	+	0.903286i	$0.358853\pi$
$984$		0			0
$985$	−840.000	+	1494.79i	−0.852792	+	1.51755i
$986$		0			0
$987$		−	279.242i		−	0.282920i
$988$		0			0
$989$		−	601.528i		−	0.608218i
$990$		0			0
$991$		−	208.000i		−	0.209889i	−0.994478	−	0.104945i	$-0.966534\pi$
$991$		−	208.000i		−	0.209889i	0.994478	−	0.104945i	$-0.0334665\pi$
$992$		0			0
$993$			704.687i			0.709655i
$994$		0			0
$995$	−819.473	−	460.504i	−0.823591	−	0.462818i
$996$		0			0
$997$	950.402			0.953262			0.476631	−	0.879104i	$-0.341858\pi$
$997$	950.402			0.953262			0.476631	+	0.879104i	$0.341858\pi$
$998$		0			0
$999$		−	432.000i		−	0.432432i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.3.e.a.159.7	yes	8
4.3	odd	2	inner	320.3.e.a.159.3	yes	8
5.2	odd	4		1600.3.g.h.351.5		8
5.3	odd	4		1600.3.g.h.351.3		8
5.4	even	2	inner	320.3.e.a.159.1	✓	8
8.3	odd	2	inner	320.3.e.a.159.6	yes	8
8.5	even	2	inner	320.3.e.a.159.2	yes	8
16.3	odd	4		1280.3.h.i.1279.5		8
16.5	even	4		1280.3.h.i.1279.8		8
16.11	odd	4		1280.3.h.i.1279.4		8
16.13	even	4		1280.3.h.i.1279.1		8
20.3	even	4		1600.3.g.h.351.6		8
20.7	even	4		1600.3.g.h.351.4		8
20.19	odd	2	inner	320.3.e.a.159.5	yes	8
40.3	even	4		1600.3.g.h.351.1		8
40.13	odd	4		1600.3.g.h.351.8		8
40.19	odd	2	inner	320.3.e.a.159.4	yes	8
40.27	even	4		1600.3.g.h.351.7		8
40.29	even	2	inner	320.3.e.a.159.8	yes	8
40.37	odd	4		1600.3.g.h.351.2		8
80.19	odd	4		1280.3.h.i.1279.2		8
80.29	even	4		1280.3.h.i.1279.6		8
80.59	odd	4		1280.3.h.i.1279.7		8
80.69	even	4		1280.3.h.i.1279.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.e.a.159.1	✓	8	5.4	even	2	inner
320.3.e.a.159.2	yes	8	8.5	even	2	inner
320.3.e.a.159.3	yes	8	4.3	odd	2	inner
320.3.e.a.159.4	yes	8	40.19	odd	2	inner
320.3.e.a.159.5	yes	8	20.19	odd	2	inner
320.3.e.a.159.6	yes	8	8.3	odd	2	inner
320.3.e.a.159.7	yes	8	1.1	even	1	trivial
320.3.e.a.159.8	yes	8	40.29	even	2	inner
1280.3.h.i.1279.1		8	16.13	even	4
1280.3.h.i.1279.2		8	80.19	odd	4
1280.3.h.i.1279.3		8	80.69	even	4
1280.3.h.i.1279.4		8	16.11	odd	4
1280.3.h.i.1279.5		8	16.3	odd	4
1280.3.h.i.1279.6		8	80.29	even	4
1280.3.h.i.1279.7		8	80.59	odd	4
1280.3.h.i.1279.8		8	16.5	even	4
1600.3.g.h.351.1		8	40.3	even	4
1600.3.g.h.351.2		8	40.37	odd	4
1600.3.g.h.351.3		8	5.3	odd	4
1600.3.g.h.351.4		8	20.7	even	4
1600.3.g.h.351.5		8	5.2	odd	4
1600.3.g.h.351.6		8	20.3	even	4
1600.3.g.h.351.7		8	40.27	even	4
1600.3.g.h.351.8		8	40.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{3} +(2.44949 - 4.35890i) q^{5} -10.6771 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+2.44949i q^{3} +(2.44949 - 4.35890i) q^{5} -10.6771 q^{7} +3.00000 q^{9} -8.71780 q^{11} -19.5959 q^{13} +(10.6771 + 6.00000i) q^{15} +21.3542i q^{17} -26.1534 q^{19} -26.1534i q^{21} +10.6771 q^{23} +(-13.0000 - 21.3542i) q^{25} +29.3939i q^{27} -34.8712i q^{29} +4.00000i q^{31} -21.3542i q^{33} +(-26.1534 + 46.5403i) q^{35} -14.6969 q^{37} -48.0000i q^{39} -24.0000 q^{41} -56.3383i q^{43} +(7.34847 - 13.0767i) q^{45} +10.6771 q^{47} +65.0000 q^{49} -52.3068 q^{51} -48.9898 q^{53} +(-21.3542 + 38.0000i) q^{55} -64.0625i q^{57} +43.5890 q^{59} +26.1534i q^{61} -32.0312 q^{63} +(-48.0000 + 85.4166i) q^{65} +7.34847i q^{67} +26.1534i q^{69} +84.0000i q^{71} +106.771i q^{73} +(52.3068 - 31.8434i) q^{75} +93.0806 q^{77} -100.000i q^{79} -45.0000 q^{81} +17.1464i q^{83} +(93.0806 + 52.3068i) q^{85} +85.4166 q^{87} +150.000 q^{89} +209.227 q^{91} -9.79796 q^{93} +(-64.0625 + 114.000i) q^{95} -21.3542i q^{97} -26.1534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 104 q^{25} - 192 q^{41} + 520 q^{49} - 384 q^{65} - 360 q^{81} + 1200 q^{89}+O(q^{100}) \) 8 * q + 24 * q^9 - 104 * q^25 - 192 * q^41 + 520 * q^49 - 384 * q^65 - 360 * q^81 + 1200 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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